Many areas of signal processing utilize so-called eigendecompositions of covariance matrices. Such decompositions generally provide a basis, commonly known as a Karhunen-Loève (KL) basis, in which signal expansion coefficients are uncorrelated. The KL basis or a signal represented in the KL basis can immediately reveal crucial information in many applications. For example, in an array signal processing application, the KL basis is used to estimate directions of arrival of plane waves. This application allows the received signal to be separated into a signal subspace and a noise subspace, and is therefore called a subspace method. The signal space often changes with time, necessitating the use of subspace tracking techniques.
Transform coding is another area in which it is useful to track a KL basis. In transform coding, a KL basis gives the best representation for encoding a Gaussian vector with scalar quantization and scalar entropy coding. When a KL basis is not known in advance or varies slowly, adaptive estimation or tracking can enhance performance.
Gradient methods for adaptive signal-subspace and noise-subspace estimation are described in J.-F. Yang et al., “Adaptive Eigensubspace Algorithms for Direction or Frequency Estimation and Tracking,” IEEE Trans. Acoust. Speech Signal Proc., Vol. 36, No. 2, pp. 241–251, February 1988. However, these methods are complicated both numerically and theoretically by orthonormalization steps. Adaptation of a Givens parameterized transform, which eliminates the need for explicit orthonormalization, was suggested as a method for finding a Karhunen-Loève transform in P. A. Regalia, “An Adaptive Unit Norm Filter With Applications to Signal Analysis and Karhunen-Loève Transformations,” IEEE Trans. Circuits and Systems, Vol. 37, No. 5, pp. 646–649, May 1990. However, this approach fails to provide suitable convergence results. Subsequent work either does not address step size selection, e.g., P. A. Regalia et al., “Rational Subspace Estimation Using Adaptive Lossless Filters,” IEEE Trans. Signal Proc., Vol. 40, No. 10, pp. 2392–2405, October 1992, or considers only the most rigorous form of convergence in which step sizes must shrink to zero, gradually but not too quickly, e.g., J.-P. Delmas, “Performances Analysis of Parameterized Adaptive Eigensubspace Algorithms,” Proc. IEEE Int. Conf. Acoust., Speech and Signal Proc., Detroit, Mich., pp. 2056–2059, May 1995, J.-P. Delmas, “Adaptive Harmonic Jammer Canceler,” IEEE Trans. Signal Proc., Vol. 43, No. 10, pp. 2323–2331, October 1995, and J.-P. Delmas, “Performances Analysis of a Givens Parameterized Adaptive Eigenspace Algorithm,” Signal Proc., Vol. 68, No. 1, pp. 87–105, July 1998. The latter form of adaptation, with step sizes approaching zero, is generally not suitable for use in practical applications.
For details on other conventional basis tracking techniques, see J.-F. Yang et al., “Adaptive High-Resolution Algorithms For Tracking Nonstationery Sources Without the Estimation of Source Number,” IEEE Trans. Signal Proc., Vol. 42, pp. 563–571, March 1994, P. A. Regalia, “An Unbiased Equation Error Identifier and Reduced-Order Approximations,” IEEE Trans. Signal Proc., Vol. 42, No. 6, pp. 1397–1412, June 1994, B. Champagne, “Adaptive Eigendecomposition of Data Covariance Matrices Based on First-Order Perturbations,” IEEE Trans. Signal Proc., Vol. 42, No. 10, pp. 2758–2770, October 1994, B. Yang, “Projection Approximation Subspace Tracking,” IEEE Trans. Signal Proc., Vol. 43, No. 1, pp. 95–107, January 1995, and B. Champagne et al., “Plane Rotation-Based EVD Updating Schemes For Efficient Subspace Tracking,” IEEE Trans. Signal Proc., Vol. 46, No. 7, pp. 1886–1900, July 1998.
The above-identified conventional techniques for tracking a KL basis fail to provide adequate performance, such as local convergence within specified step size bounds. In addition to exhibiting a lack of suitable convergence guarantees, most conventional techniques are computationally complicated and not highly parallelizable. A need therefore exists for improved techniques for tracking a KL basis.